Global Truncation Error

Global truncation error (GTE) is the total accumulated error after integrating over the entire interval, and it determines the Order of a method.

For a method with Local truncation error $O(d{t}^{p+1})$ per step, integrating over $N=T/dt$ steps gives

$$E_{\text{global}}\approx N\cdot O(d{t}^{p+1})=\frac{T}{dt}\cdot O(d{t}^{p+1})=O(d{t}^p)$$

so the method is order $p$.

Method	LTE per step	GTE (order)
Explicit Euler method	$O(d{t}^2)$	$O(dt)$ - first order
Midpoint method, Ralston method, Implicit Trapezoid Method	$O(d{t}^3)$	$O(d{t}^2)$ - second order
Fourth order Runge-Kutta	$O(d{t}^5)$	$O(d{t}^4)$ - fourth order

The GTE is what we measure in practice by comparing numerical solutions at different step sizes.

Local truncation error | Order of a method | Order of convergence | Convergence